%matplotlib inline
        import torch
        from d2l import torch as d2l
        
        torch.set_printoptions(2)  # 精简输出精度
        

#@save
        def multibox_prior(data, sizes, ratios):
            """生成以每个像素为中心具有不同形状的锚框"""
            in_height, in_width = data.shape[-2:]
            device, num_sizes, num_ratios = data.device, len(sizes), len(ratios)
            boxes_per_pixel = (num_sizes + num_ratios - 1)
            size_tensor = torch.tensor(sizes, device=device)
            ratio_tensor = torch.tensor(ratios, device=device)
        
            # 为了将锚点移动到像素的中心，需要设置偏移量。
            # 因为一个像素的高为1且宽为1，我们选择偏移我们的中心0.5
            offset_h, offset_w = 0.5, 0.5
            steps_h = 1.0 / in_height  # 在y轴上缩放步长
            steps_w = 1.0 / in_width  # 在x轴上缩放步长
        
            # 生成锚框的所有中心点
            center_h = (torch.arange(in_height, device=device) + offset_h) * steps_h
            center_w = (torch.arange(in_width, device=device) + offset_w) * steps_w
            shift_y, shift_x = torch.meshgrid(center_h, center_w, indexing='ij')
            shift_y, shift_x = shift_y.reshape(-1), shift_x.reshape(-1)
        
            # 生成“boxes_per_pixel”个高和宽，
            # 之后用于创建锚框的四角坐标(xmin,xmax,ymin,ymax)
            w = torch.cat((size_tensor * torch.sqrt(ratio_tensor[0]),
                           sizes[0] * torch.sqrt(ratio_tensor[1:])))\
                           * in_height / in_width  # 处理矩形输入
            h = torch.cat((size_tensor / torch.sqrt(ratio_tensor[0]),
                           sizes[0] / torch.sqrt(ratio_tensor[1:])))
            # 除以2来获得半高和半宽
            anchor_manipulations = torch.stack((-w, -h, w, h)).T.repeat(
                                                in_height * in_width, 1) / 2
        
            # 每个中心点都将有“boxes_per_pixel”个锚框，
            # 所以生成含所有锚框中心的网格，重复了“boxes_per_pixel”次
            out_grid = torch.stack([shift_x, shift_y, shift_x, shift_y],
                        dim=1).repeat_interleave(boxes_per_pixel, dim=0)
            output = out_grid + anchor_manipulations
            return output.unsqueeze(0)
        

#@save
        def show_bboxes(axes, bboxes, labels=None, colors=None):
            """显示所有边界框"""
            def _make_list(obj, default_values=None):
                if obj is None:
                    obj = default_values
                elif not isinstance(obj, (list, tuple)):
                    obj = [obj]
                return obj
        
            labels = _make_list(labels)
            colors = _make_list(colors, ['b', 'g', 'r', 'm', 'c'])
            for i, bbox in enumerate(bboxes):
                color = colors[i % len(colors)]
                rect = d2l.bbox_to_rect(bbox.detach().numpy(), color)
                axes.add_patch(rect)
                if labels and len(labels) > i:
                    text_color = 'k' if color == 'w' else 'w'
                    axes.text(rect.xy[0], rect.xy[1], labels[i],
                              va='center', ha='center', fontsize=9, color=text_color,
                              bbox=dict(facecolor=color, lw=0))
        

#@save
        def box_iou(boxes1, boxes2):
            """计算两个锚框或边界框列表中成对的交并比"""
            box_area = lambda boxes: ((boxes[:, 2] - boxes[:, 0]) *
                                      (boxes[:, 3] - boxes[:, 1]))
            # boxes1,boxes2,areas1,areas2的形状:
            # boxes1：(boxes1的数量,4),
            # boxes2：(boxes2的数量,4),
            # areas1：(boxes1的数量,),
            # areas2：(boxes2的数量,)
            areas1 = box_area(boxes1)
            areas2 = box_area(boxes2)
            # inter_upperlefts,inter_lowerrights,inters的形状:
            # (boxes1的数量,boxes2的数量,2)
            inter_upperlefts = torch.max(boxes1[:, None, :2], boxes2[:, :2])
            inter_lowerrights = torch.min(boxes1[:, None, 2:], boxes2[:, 2:])
            inters = (inter_lowerrights - inter_upperlefts).clamp(min=0)
            # inter_areasandunion_areas的形状:(boxes1的数量,boxes2的数量)
            inter_areas = inters[:, :, 0] * inters[:, :, 1]
            union_areas = areas1[:, None] + areas2 - inter_areas
            return inter_areas / union_areas
        

#@save
        def assign_anchor_to_bbox(ground_truth, anchors, device, iou_threshold=0.5):
            """将最接近的真实边界框分配给锚框"""
            num_anchors, num_gt_boxes = anchors.shape[0], ground_truth.shape[0]
            # 位于第i行和第j列的元素x_ij是锚框i和真实边界框j的IoU
            jaccard = box_iou(anchors, ground_truth)
            # 对于每个锚框，分配的真实边界框的张量
            anchors_bbox_map = torch.full((num_anchors,), -1, dtype=torch.long,
                                          device=device)
            # 根据阈值，决定是否分配真实边界框
            max_ious, indices = torch.max(jaccard, dim=1)
            anc_i = torch.nonzero(max_ious >= iou_threshold).reshape(-1)
            box_j = indices[max_ious >= iou_threshold]
            anchors_bbox_map[anc_i] = box_j
            col_discard = torch.full((num_anchors,), -1)
            row_discard = torch.full((num_gt_boxes,), -1)
            for _ in range(num_gt_boxes):
                max_idx = torch.argmax(jaccard)
                box_idx = (max_idx % num_gt_boxes).long()
                anc_idx = (max_idx / num_gt_boxes).long()
                anchors_bbox_map[anc_idx] = box_idx
                jaccard[:, box_idx] = col_discard
                jaccard[anc_idx, :] = row_discard
            return anchors_bbox_map
        

#@save
        def offset_boxes(anchors, assigned_bb, eps=1e-6):
            """对锚框偏移量的转换"""
            c_anc = d2l.box_corner_to_center(anchors)
            c_assigned_bb = d2l.box_corner_to_center(assigned_bb)
            offset_xy = 10 * (c_assigned_bb[:, :2] - c_anc[:, :2]) / c_anc[:, 2:]
            offset_wh = 5 * torch.log(eps + c_assigned_bb[:, 2:] / c_anc[:, 2:])
            offset = torch.cat([offset_xy, offset_wh], axis=1)
            return offset
        

#@save
        def multibox_target(anchors, labels):
            """使用真实边界框标记锚框"""
            batch_size, anchors = labels.shape[0], anchors.squeeze(0)
            batch_offset, batch_mask, batch_class_labels = [], [], []
            device, num_anchors = anchors.device, anchors.shape[0]
            for i in range(batch_size):
                label = labels[i, :, :]
                anchors_bbox_map = assign_anchor_to_bbox(
                    label[:, 1:], anchors, device)
                bbox_mask = ((anchors_bbox_map >= 0).float().unsqueeze(-1)).repeat(
                    1, 4)
                # 将类标签和分配的边界框坐标初始化为零
                class_labels = torch.zeros(num_anchors, dtype=torch.long,
                                           device=device)
                assigned_bb = torch.zeros((num_anchors, 4), dtype=torch.float32,
                                          device=device)
                # 使用真实边界框来标记锚框的类别。
                # 如果一个锚框没有被分配，标记其为背景（值为零）
                indices_true = torch.nonzero(anchors_bbox_map >= 0)
                bb_idx = anchors_bbox_map[indices_true]
                class_labels[indices_true] = label[bb_idx, 0].long() + 1
                assigned_bb[indices_true] = label[bb_idx, 1:]
                # 偏移量转换
                offset = offset_boxes(anchors, assigned_bb) * bbox_mask
                batch_offset.append(offset.reshape(-1))
                batch_mask.append(bbox_mask.reshape(-1))
                batch_class_labels.append(class_labels)
            bbox_offset = torch.stack(batch_offset)
            bbox_mask = torch.stack(batch_mask)
            class_labels = torch.stack(batch_class_labels)
            return (bbox_offset, bbox_mask, class_labels)
        

labels = multibox_target(anchors.unsqueeze(dim=0),
                                 ground_truth.unsqueeze(dim=0))
        

#@save
        def offset_inverse(anchors, offset_preds):
            """根据带有预测偏移量的锚框来预测边界框"""
            anc = d2l.box_corner_to_center(anchors)
            pred_bbox_xy = (offset_preds[:, :2] * anc[:, 2:] / 10) + anc[:, :2]
            pred_bbox_wh = torch.exp(offset_preds[:, 2:] / 5) * anc[:, 2:]
            pred_bbox = torch.cat((pred_bbox_xy, pred_bbox_wh), axis=1)
            predicted_bbox = d2l.box_center_to_corner(pred_bbox)
            return predicted_bbox
        

#@save
        def nms(boxes, scores, iou_threshold):
            """对预测边界框的置信度进行排序"""
            B = torch.argsort(scores, dim=-1, descending=True)
            keep = []  # 保留预测边界框的指标
            while B.numel() > 0:
                i = B[0]
                keep.append(i)
                if B.numel() == 1: break
                iou = box_iou(boxes[i, :].reshape(-1, 4),
                              boxes[B[1:], :].reshape(-1, 4)).reshape(-1)
                inds = torch.nonzero(iou <= iou_threshold).reshape(-1)
                B = B[inds + 1]
            return torch.tensor(keep, device=boxes.device)
        

#@save
        def multibox_detection(cls_probs, offset_preds, anchors, nms_threshold=0.5,
                               pos_threshold=0.009999999):
            """使用非极大值抑制来预测边界框"""
            device, batch_size = cls_probs.device, cls_probs.shape[0]
            anchors = anchors.squeeze(0)
            num_classes, num_anchors = cls_probs.shape[1], cls_probs.shape[2]
            out = []
            for i in range(batch_size):
                cls_prob, offset_pred = cls_probs[i], offset_preds[i].reshape(-1, 4)
                conf, class_id = torch.max(cls_prob[1:], 0)
                predicted_bb = offset_inverse(anchors, offset_pred)
                keep = nms(predicted_bb, conf, nms_threshold)
        
                # 找到所有的non_keep索引，并将类设置为背景
                all_idx = torch.arange(num_anchors, dtype=torch.long, device=device)
                combined = torch.cat((keep, all_idx))
                uniques, counts = combined.unique(return_counts=True)
                non_keep = uniques[counts == 1]
                all_id_sorted = torch.cat((keep, non_keep))
                class_id[non_keep] = -1
                class_id = class_id[all_id_sorted]
                conf, predicted_bb = conf[all_id_sorted], predicted_bb[all_id_sorted]
                # pos_threshold是一个用于非背景预测的阈值
                below_min_idx = (conf < pos_threshold)
                class_id[below_min_idx] = -1
                conf[below_min_idx] = 1 - conf[below_min_idx]
                pred_info = torch.cat((class_id.unsqueeze(1),
                                       conf.unsqueeze(1),
                                       predicted_bb), dim=1)
                out.append(pred_info)
            return torch.stack(out)
        

anchors = torch.tensor([\[0.1, 0.08, 0.52, 0.92], [0.08, 0.2, 0.56, 0.95],
                              [0.15, 0.3, 0.62, 0.91], [0.55, 0.2, 0.9, 0.88]\])
        offset_preds = torch.tensor([0] * anchors.numel())
        cls_probs = torch.tensor([\[0] * 4,  # 背景的预测概率
                              [0.9, 0.8, 0.7, 0.1],  # 狗的预测概率
                              [0.1, 0.2, 0.3, 0.9]\])  # 猫的预测概率